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Yicheng Pang - One of the best experts on this subject based on the ideXlab platform.
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delta shock wave to the compressible fluid flow with the generalized chaplygin gas
International Journal of Non-linear Mechanics, 2018Co-Authors: Yicheng PangAbstract:Abstract We concern with the Riemann problem the compressible fluid flow with the generalized Chaplygin gas. With the analysis on the phase plane, we rigorously confirm the occurrence of delta shock wave with Dirac delta function in density. Then the formation mechanism, generalized Rankine–Hugoniot relation and Entropy Condition for the delta shock wave are clarified. Based on these preparations, five kinds of exact solutions are obtained. Finally, the corresponding numerical results are also presented to illustrate our analysis.
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Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas
SpringerOpen, 2018Co-Authors: Yicheng Pang, Jinhuan WangAbstract:Abstract The Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas is considered. We obtain two kinds of exact solutions. The first one consists of contact discontinuities, while the other one involves a delta shock wave in which both density and internal energy contain a Dirac delta function. The position, speed and weights of the delta shock wave are derived from both generalized Rankine–Hugoniot relation and Entropy Condition, which are established in detail. Moreover, the solutions are no longer self-similar due to the influence of the constant external force
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delta shock wave in the compressible euler equations for a chaplygin gas
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Yicheng PangAbstract:Abstract We study the Riemann problem of the compressible Euler equations for the Chaplygin gas. With the analysis on the physically relevant region, we obtain two kinds of Riemann solutions by using the method of characteristic analysis. One composes of three contact discontinuities, and the other involves a delta shock wave in which both density and internal energy contain Dirac delta function simultaneously. We propose both generalized Rankine–Hugoniot relation and Entropy Condition for this type of delta shock wave. The numerical results coinciding with the theoretical analysis are also presented.
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delta shock wave with dirac delta function in multiple components for the system of generalized chaplygin gas dynamics
Boundary Value Problems, 2016Co-Authors: Yicheng PangAbstract:We study the Riemann problem for the compressible Euler equations with the generalized Chaplygin gas. Based on the analysis on the physically relevant region, we obtain five kinds of exact solutions. It is shown that a delta shock wave with Dirac delta function in both density and internal energy develops in the exact solutions. The formation mechanism, generalized Rankine-Hugoniot relation and Entropy Condition are clarified for this type of delta shock wave. The numerical results are also presented to confirm this type of delta shock wave.
Huicheng Yin - One of the best experts on this subject based on the ideXlab platform.
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on the instability problem of a 3 d transonic oblique shock wave
Advances in Mathematics, 2015Co-Authors: Huicheng YinAbstract:Abstract In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated on p. 317 of [7] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine–Hugoniot Conditions and the physical Entropy Condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined with respect to the periodic perturbation, which implies that the 3-D transonic shock is unstable in general.
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On the instability problem of a 3-D transonic oblique shock wave
2014Co-Authors: Li Liang, Xu Gang, Huicheng YinAbstract:In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated in pages 317 of [9] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot Conditions and the physical Entropy Condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined, which implies that the 3-D transonic shock is unstable in general.Comment: 65 page
Pinto, Antonio Carlos Capeleiro - One of the best experts on this subject based on the ideXlab platform.
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Esquemas de alta resolução para controle da dispersão numerica em simulação de reservatorios
[s.n.], 2018Co-Authors: Pinto, Antonio Carlos CapeleiroAbstract:Orientador: Antonio Claudio de França CorreaDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecanicaResumo: O esquema tradicional de simulação de reservatórios por diferenças finitas utiliza o método de ponderação a montante para aproximar os componentes do termo de fluxo convectivo nas interfaces entre os blocos. Esse procedimento estabiliza a solução numérica, mas introduz graves erros de dispersão numérica, dificultando a correta interpretação dos resultados simulados. Os esquemas exponenciais são alternativas razoáveis quando o termo difusivo domina. No entanto, conforme demonstramos neste trabalho, tendem para o método de ponderação a montante quando o fluxo é muito convectivo. Os métodos de diferenças finitas de ordem mais alta, como o esquema de Leonard, reduzem a dispersão numérica, mas podem produzir soluções fisicamente incorretas quando a equação de conservação assume a forma hiperbólica. Demonstramos, através da solução de algumas equações não-lineares clássicas, que isto ocorre porque a condição de entropia não é obedecida. A condição de entropia é um critério matemático que permite selecionar a solução correta entre as soluções fracas do problema. Os esquemas de Diminuição das Variações Totais (TVD) possuem a notável propriedade de produzirem soluções de alta resolução que obedecem ao princípio da entropia e, conseqüentemente, são fisicamente corretas. Apresentamos uma comparação do desempenho de diversos métodos de diferenças finitas para resolver alguns problemas de engenharia de reservatórios, como: equação da convecção-difusão em lD, equação de Buckley-Leverett e fluxo de traçador em 2D. Os esquemas TVD foram também implementados em um modelo "black-oil" bifásico, e os resultados são discutidos para várias formulações, como IMPES, semi-implícito e totalmente implícito. InclUÍmos no modelo a opção de injeção de traçadores na fase água, considerando o tensor dispersão completo e adsorção com a rocha. São feitas comparações com as soluções obtidas com o método de ponderação a montante e malha refinada e, sempre que possível, com soluções analíticas. O efeito de orientação de malha é estudado. São incluídos diversos exemplos práticosAbstract: Standard finite-diference reservoir simulation normally uses single-point upstream welghting to approximate the components of the convective flow term at the interfaces between blocks. This procedure stabilizes the numerical solution, but introduces high levels of numerical dispersion, difficulting the correct interpretation of the simulated results. Exponential schemes are reasonable when diffusion dominates, but, as we show in this work, reduce to single-point uspstream when the flow is toa convective. Higher-order finite-difference methods, like Leonard's scheme, are, in general, able to reduce the numerical dispersion, but may produce non-physical solutions when the conservation equation assumes a hyperbolic formo We demonstrate, through the numerical solution of some classical non-linear equations, that this occurs because the Entropy Condition is violated. The Entropy Condition is a mathematical cri teria to select the correct solution among weak solutions of the problem. Total Variation Diminishing (TVD) methods have the remarkable property of producing high resolution solutions which obey the Entropy Condition, and, consequently, are physically consistent. A comparison of the performance of the various methods is presented for some reservoir engineering problems: lD convection-diffusion equation, Buckley-Leverett equation and 2D single-phase tracer flow. TVD schemes were also implemented on a two-phase black-oil model, and resuIts are discussed for various formuIations, such as IMPES, semi-implicit and fully-implicit. We also included options of tracer injection in the water phase, considering fuIl dispersion tensor and adsorption with the rock. Comparisons are made with refined grid single-point upstream solutions and, whenever possible, with analytical solutions. Grid orientation effect is investigated. Practical examples are also included.MestradoMestre em Engenharia de Petróle
Antonio Carlos Capeleiro Pinto - One of the best experts on this subject based on the ideXlab platform.
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Esquemas de alta resolução para controle da dispersão numerica em simulação de reservatorios
2017Co-Authors: Antonio Carlos Capeleiro PintoAbstract:Resumo: O esquema tradicional de simulação de reservatórios por diferenças finitas utiliza o método de ponderação a montante para aproximar os componentes do termo de fluxo convectivo nas interfaces entre os blocos. Esse procedimento estabiliza a solução numérica, mas introduz graves erros de dispersão numérica, dificultando a correta interpretação dos resultados simulados. Os esquemas exponenciais são alternativas razoáveis quando o termo difusivo domina. No entanto, conforme demonstramos neste trabalho, tendem para o método de ponderação a montante quando o fluxo é muito convectivo. Os métodos de diferenças finitas de ordem mais alta, como o esquema de Leonard, reduzem a dispersão numérica, mas podem produzir soluções fisicamente incorretas quando a equação de conservação assume a forma hiperbólica. Demonstramos, através da solução de algumas equações não-lineares clássicas, que isto ocorre porque a condição de entropia não é obedecida. A condição de entropia é um critério matemático que permite selecionar a solução correta entre as soluções fracas do problema. Os esquemas de Diminuição das Variações Totais (TVD) possuem a notável propriedade de produzirem soluções de alta resolução que obedecem ao princípio da entropia e, conseqüentemente, são fisicamente corretas. Apresentamos uma comparação do desempenho de diversos métodos de diferenças finitas para resolver alguns problemas de engenharia de reservatórios, como: equação da convecção-difusão em lD, equação de Buckley-Leverett e fluxo de traçador em 2D. Os esquemas TVD foram também implementados em um modelo "black-oil" bifásico, e os resultados são discutidos para várias formulações, como IMPES, semi-implícito e totalmente implícito. InclUÍmos no modelo a opção de injeção de traçadores na fase água, considerando o tensor dispersão completo e adsorção com a rocha. São feitas comparações com as soluções obtidas com o método de ponderação a montante e malha refinada e, sempre que possível, com soluções analíticas. O efeito de orientação de malha é estudado. São incluídos diversos exemplos práticosAbstract: Standard finite-diference reservoir simulation normally uses single-point upstream welghting to approximate the components of the convective flow term at the interfaces between blocks. This procedure stabilizes the numerical solution, but introduces high levels of numerical dispersion, difficulting the correct interpretation of the simulated results. Exponential schemes are reasonable when diffusion dominates, but, as we show in this work, reduce to single-point uspstream when the flow is toa convective. Higher-order finite-difference methods, like Leonard's scheme, are, in general, able to reduce the numerical dispersion, but may produce non-physical solutions when the conservation equation assumes a hyperbolic formo We demonstrate, through the numerical solution of some classical non-linear equations, that this occurs because the Entropy Condition is violated. The Entropy Condition is a mathematical cri teria to select the correct solution among weak solutions of the problem. Total Variation Diminishing (TVD) methods have the remarkable property of producing high resolution solutions which obey the Entropy Condition, and, consequently, are physically consistent. A comparison of the performance of the various methods is presented for some reservoir engineering problems: lD convection-diffusion equation, Buckley-Leverett equation and 2D single-phase tracer flow. TVD schemes were also implemented on a two-phase black-oil model, and resuIts are discussed for various formuIations, such as IMPES, semi-implicit and fully-implicit. We also included options of tracer injection in the water phase, considering fuIl dispersion tensor and adsorption with the rock. Comparisons are made with refined grid single-point upstream solutions and, whenever possible, with analytical solutions. Grid orientation effect is investigated. Practical examples are also include
Hanchun Yang - One of the best experts on this subject based on the ideXlab platform.
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delta shock waves with dirac delta function in both components for systems of conservation laws
Journal of Differential Equations, 2014Co-Authors: Hanchun Yang, Yanyan ZhangAbstract:Abstract We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and Entropy Condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.
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On a Nonsymmetric Keyfitz-Kranzer System of Conservation Laws with Generalized and Modified Chaplygin Gas Pressure Law
Hindawi Limited, 2013Co-Authors: Hongjun Cheng, Hanchun YangAbstract:This paper is devoted to the study of a nonsymmetric Keyfitz-Kranzer system of conservation laws with the generalized and modified Chaplygin gas pressure law, which may admit delta shock waves, a topic of interest. Firstly, we solve the Riemann problems with piecewise constant data having a single discontinuity. For the generalized Chaplygin gas pressure law, the solution consists of three different structures: R+J, S+J, and δ. Existence and uniqueness of delta shock solution are established under the generalized Rankine-Hugoniot relation and Entropy Condition. For the modified Chaplygin gas pressure law, the structures of solution are R+J and S+J. Secondly, we discuss the limits of Riemann solutions for the modified Chaplygin gas pressure law as the pressure law tends to the generalized Chaplygin gas one. In particular, for some cases, the solution S+J tends to a delta shock wave, and it is different from the delta shock wave for the generalized Chaplygin gas pressure law with the same initial data. Thirdly, we simulate the Riemann solutions and examine the formation process of delta shock wave by employing the Nessyahu-Tadmor scheme. The numerical results are coincident with the theoretical analysis
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Riemann problem for the relativistic Chaplygin Euler equations
Journal of Mathematical Analysis and Applications, 2011Co-Authors: Hongjun Cheng, Hanchun YangAbstract:Abstract The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine–Hugoniot relation and Entropy Condition, the existence and uniqueness of delta-shock solutions are established.
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Generalized Plane Delta-Shock Waves for n-Dimensional Zero-Pressure Gas Dynamics
Journal of Mathematical Analysis and Applications, 2001Co-Authors: Hanchun YangAbstract:Abstract With the help of a generalized plane wave solution, we study a type of generalized plane delta-shock wave for the n -dimensional zero-pressure gas dynamics and refine its generalized Rankine–Hugoniot relation which is a system of ordinary equations. This relation describes accurately the character of the generalized plane delta-shock: location, propagation speed, and weight. Under a suitable Entropy Condition, four different explicit constructions of solutions for a kind of Riemann problem with Radon measure as initial data are established uniquely. The overtaking of two plane delta-shocks forming a new generalized plane delta-shock is also investigated. Finally, the 2-D Riemann problem with four pieces of initial data is solved in a simplified situation.